Inverse Response

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Inverse Response When the initial response of a dynamic system is in a direction opposite to the final outcome, it is called an inverse response.

Inverse response typically results when two separate effects are occurring at the same time, but with different directions and dynamics. The system is thus two transfer functions coupled in parallel, so the individual outputs can be added to obtain the overall response.

One example of inverse response is the behavior of boilers called shrink and swell. If the demand load on a steam drum suddenly increases, the drum pressure will drop. The water in the drum is superheated, so vapor bubbles form beneath the surface of the liquid, causing the liquid density to decrease and the apparent level of liquid in the drum to increase, even though the actual mass of material in the drum has not changed. Once the pressure disturbance has equilibrated, the liquid level will fall back to its true value.

Another example of paired effects occurs when changes are made in the vapor boilup rate at the base of a distillation column. The decrease in base holdup caused by the increased draw and the increase in stripping caused by increased vapor flow have opposite effects on the bottoms composition. This can produce an apparent inverse response.

Block Diagram: Inverse Response

To better understand inverse response, consider two first order transfer functions connected in parallel (as shown in the block diagram). We will look at the case where Process 1 has smaller time constant and gain than Process 2. Components of Inverse Response

Because Process 1 is faster, it dominates the initial response, but since Process 2 has a larger magnitude (because of its larger steady state gain), it dominates the steady state response.

In the Laplace domain, we represent the dynamics as:

G;P = 'quot(K;1,?tau?;1*s+1) - 'quot(K;2,?tau?;2*s+1)
These transfer functions have poles at _'quot(1,?tau?;1) and _'quot(1,?tau?;2) .

If we reduce the block diagram and do the algebra to get the overall transfer function, we get

G;P = 'quot(K;1*?tau?;2*s + K;1 - K;2*?tau?;1*s - K;2, (?tau?;1*s+1)*(?tau?;2*s+1))
G;P = 'quot((K;1*?tau?;2- K;2*?tau?;1)*s - (K;2-K;1), (?tau?;1*s+1)*(?tau?;2*s+1))
which is arranged so that both numerator terms, K;1*?tau?;2-K;2*?tau?;1 and (K;2-K;1) are greater than zero. This can be further rearranged to:
G;P = 'quot(_1*(K;2- K;1)*(_1*'quot(K;1*?tau?;2-K;2*?tau?;1,K;2- K;1)*s+1),(?tau?;1*s+1)*(?tau?;2*s+1))
This combined transfer function still has the same two open loop poles, at _'quot(1,?tau?;1) and _'quot(1,?tau?;2) but has acquired ann open loop zero at 'quot(K;2-K;1, K;1*?tau?;2- K;2*?tau?;1) Inverse Response

Since the poles are not changed and remain in the left half-plane, the stability and "speed" of the process are not substantially effected; however, the system displays inverse response. The indicator is the zero in the right half-plane. This is characteristic of inverse response, so if you notice a right half-plane zero, you should anticipate inverse response.

References:

  1. Luyben, W.L. Process Modeling, Simulation and Control for Chemical Engineers (2nd Edition), McGraw-Hill, 1990, pp. 398-402.
  2. Riggs, J.B., Chemical Process Control (2nd Edition), Ferret, 2001, pp. 196-97.

R.M. Price
Original: 10/25/93
Modified: 2/19/97, 6/30/2003

Copyright 1997, 2003 by R.M. Price -- All Rights Reserved

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